Multilevel preconditioning of discontinuous Galerkin spectral element methods. Part I: geometrically conforming meshes

Abstract

This paper is concerned with the design, analysis and implementation of preconditioning con- cepts for spectral DG discretizations of elliptic boundary value problems. The far term goal is to obtain robust solvers for the fully flexible case. By this we mean Discontinuous Galerkin schemes on locally re- fined quadrilateral or hexahedral partitions with hanging nodes and variable polynomial degrees that could, in principle, be arbitrarily large only subject to some weak grading constraints. In this paper, as a first step, we focus on varying arbitrarily large degrees while keeping the mesh geometrically conforming since this will be seen to exhibit already some essential obstructions. The conceptual foundation of the envisaged preconditioners is the auxiliary space method, or in fact, an iterated variant of it. The main conceptual pillars that will be shown in this framework to yield optimal preconditioners are Legendre-Gaus-Lobatto grids in connection with certain associated anisotropic nested dyadic grids. Here optimal means that the preconditioned systems exhibit uniformly bounded condition numbers. Moreover, the preconditioners have a modular form that facilitates somewhat simplified partial realizations at the expense of a moderate loss of eciency. Our analysis is complemented by careful quantitative experimental studies of the main components.